Found problems: 85335
2016 Greece Junior Math Olympiad, 1
If $n$ is positive integer and $p, q, r$ are primes solve the system: $pqr=n$ and $(p+1)(q+1)r=n+138$
2024 China Western Mathematical Olympiad, 8
Given a positive integer $n \geq 2$. Let $a_{ij}$ $(1 \leq i,j \leq n)$ be $n^2$ non-negative reals and their sum is $1$. For $1\leq i \leq n$, define $R_i=max_{1\leq k \leq n}(a_{ik})$. For $1\leq j \leq n$, define $C_j=min_{1\leq k \leq n}(a_{kj})$
Find the maximum value of $C_1C_2 \cdots C_n(R_1+R_2+ \cdots +R_n)$
2011 Benelux, 3
If $k$ is an integer, let $\mathrm{c}(k)$ denote the largest cube that is less than or equal to $k$. Find all positive integers $p$ for which the following sequence is bounded:
$a_0 = p$ and $a_{n+1} = 3a_n-2\mathrm{c}(a_n)$ for $n \geqslant 0$.
1987 National High School Mathematics League, 2
For a rhombus with side length of 5, length of one of its diagonal is not larger than $6$, length of the other diagonal is not smaller than $6$, then the maximum value of the sum of the two diagonals is
$\text{(A)}10\sqrt{2}\qquad\text{(B)}14\qquad\text{(C)}5\sqrt{6}\qquad\text{(D)}12$
2000 Irish Math Olympiad, 3
Let $ f(x)\equal{}5x^{13}\plus{}13x^5\plus{}9ax$. Find the least positive integer $ a$ such that $ 65$ divides $ f(x)$ for every integer $ x$.
2024 JBMO TST - Turkey, 2
A real number is written on each square of a $2024 \times 2024$ chessboard. It is given that the sum of all real numbers on the board is $2024$. Then, the board is covered by $1 \times 2$ or $2\times 1$ dominos such that there isn't any square that is covered by two different dominoes. For each domino, Aslı deletes $2$ numbers covered by it and writes $0$ on one of the squares and the sum of the two numbers on the other square. Find the maximum number $k$ such that after Aslı finishes her moves, there exists a column or row where the sum of all the numbers on it is at least $k$ regardless of how dominos were replaced and the real numbers were written initially.
2011 Belarus Team Selection Test, 1
Find the least possible number of elements which can be deleted from the set $\{1,2,...,20\}$ so that the sum of no two different remaining numbers is not a perfect square.
N. Sedrakian , I.Voronovich
2020 Argentina National Olympiad Level 2, 1
Fede chooses $50$ distinct integers from the set $\{1, 2, 3, \ldots, 100\}$ such that their sum equals $2900$. Determine the minimum number of even numbers that can be among the $50$ numbers chosen by Fede.
2010 Kazakhstan National Olympiad, 5
Let $n \geq 2$ be an integer. Define $x_i =1$ or $-1$ for every $i=1,2,3,\cdots, n$.
Call an operation [i]adhesion[/i], if it changes the string $(x_1,x_2,\cdots,x_n)$ to $(x_1x_2, x_2x_3, \cdots ,x_{n-1}x_n, x_nx_1)$ .
Find all integers $n \geq 2$ such that the string $(x_1,x_2,\cdots, x_n)$ changes to $(1,1,\cdots,1)$ after finitely [i]adhesion[/i] operations.
2021 Balkan MO Shortlist, A6
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$f(xy) = f(x)f(y) + f(f(x + y))$$
holds for all $x, y \in \mathbb{R}$.
2010 Romania National Olympiad, 3
Let $VABCD$ be a regular pyramid, having the square base $ABCD$. Suppose that on the line $AC$ lies a point $M$ such that $VM=MB$ and $(VMB)\perp (VAB)$. Prove that $4AM=3AC$.
[i]Mircea Fianu[/i]
2013 ELMO Shortlist, 2
Let $ABC$ be a scalene triangle with circumcircle $\Gamma$, and let $D$,$E$,$F$ be the points where its incircle meets $BC$, $AC$, $AB$ respectively. Let the circumcircles of $\triangle AEF$, $\triangle BFD$, and $\triangle CDE$ meet $\Gamma$ a second time at $X,Y,Z$ respectively. Prove that the perpendiculars from $A,B,C$ to $AX,BY,CZ$ respectively are concurrent.
[i]Proposed by Michael Kural[/i]
2017 Iran MO (3rd round), 3
Let $k$ be a positive integer. Find all functions $f:\mathbb{N}\to \mathbb{N}$ satisfying the following two conditions:\\
• For infinitely many prime numbers $p$ there exists a positve integer $c$ such that $f(c)=p^k$.\\
• For all positive integers $m$ and $n$, $f(m)+f(n)$ divides $f(m+n)$.
2020 Korea Junior Math Olympiad, 3
The permutation $\sigma$ consisting of four words $A,B,C,D$ has $f_{AB}(\sigma)$, the sum of the number of $B$ placed rightside of every $A$. We can define $f_{BC}(\sigma)$,$f_{CD}(\sigma)$,$f_{DA}(\sigma)$ as the same way too.
For example, $\sigma=ACBDBACDCBAD$, $f_{AB}(\sigma)=3+1+0=4$, $f_{BC}(\sigma)=4$,$f_{CD}(\sigma)=6$, $f_{DA}(\sigma)=3$
Find the maximal value of $f_{AB}(\sigma)+f_{BC}(\sigma)+f_{CD}(\sigma)+f_{DA}(\sigma)$, when $\sigma$ consists of $2020$ letters for each $A,B,C,D$
2015 Indonesia MO Shortlist, N1
A triple integer $(a, b, c)$ is called [i]brilliant [/i] when it satisfies:
(i) $a> b> c$ are prime numbers
(ii) $a = b + 2c$
(iii) $a + b + c$ is a perfect square number
Find the minimum value of $abc$ if triple $(a, b, c)$ is [i]brilliant[/i].
1966 IMO Longlists, 40
For a positive real number $p$, find all real solutions to the equation
\[\sqrt{x^2 + 2px - p^2} -\sqrt{x^2 - 2px - p^2} =1.\]
2019 AMC 12/AHSME, 5
Two lines with slopes $\dfrac{1}{2}$ and $2$ intersect at $(2,2)$. What is the area of the triangle enclosed by these two lines and the line $x+y=10 ?$
$\textbf{(A) } 4 \qquad\textbf{(B) } 4\sqrt{2} \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 6\sqrt{2}$
2018 BAMO, E/3
Suppose that $2002$ numbers, each equal to $1$ or $-1$, are written around a circle. For every two adjacent numbers, their product is taken; it turns out that the sum of all $2002$ such products is negative. Prove that the sum of the original numbers has absolute value less than or equal to $1000$. (The absolute value of $x$ is usually denoted by $|x|$. It is equal to $x$ if $x \ge 0$, and to $-x$ if $x < 0$. For example, $|6| = 6, |0| = 0$, and $|-7| = 7$.)
2007 ITest, 56
Let $T=\text{TNFTPP}$. In the binary expansion of \[\dfrac{2^{2007}-1}{2^T-1},\] how many of the first $10,000$ digits to the right of the radix point are $0$'s?
1983 AMC 12/AHSME, 7
Alice sells an item at $\$10$ less than the list price and receives $10\%$ of her selling price as her commission. Bob sells the same item at $\$20$ less than the list price and receives $20\%$ of his selling price as his commission. If they both get the same commission, then the list price is
$ \textbf{(A)}\ \$20\qquad\textbf{(B)}\ \$30\qquad\textbf{(C)}\ \$50\qquad\textbf{(D)}\ \$70\qquad\textbf{(E)}\ \$100 $
2006 Pan African, 1
Let $AB$ and $CD$ be two perpendicular diameters of a circle with centre $O$. Consider a point $M$ on the diameter $AB$, different from $A$ and $B$. The line $CM$ cuts the circle again at $N$. The tangent at $N$ to the circle and the perpendicular at $M$ to $AM$ intersect at $P$. Show that $OP = CM$.
2015 Iran MO (3rd round), 2
Let $ABC$ be a triangle with orthocenter $H$ and circumcenter $O$. Let $K$ be the midpoint of $AH$. point $P$ lies on $AC$ such that $\angle BKP=90^{\circ}$. Prove that $OP\parallel BC$.
2005 District Olympiad, 3
Let $ABC$ be a non-right-angled triangle and let $H$ be its orthocenter. Let $M_1,M_2,M_3$ be the midpoints of the sides $BC$, $CA$, $AB$ respectively. Let $A_1$, $B_1$, $C_1$ be the symmetrical points of $H$ with respect to $M_1$, $M_2$ and $M_3$ respectively, and let $A_2$, $B_2$, $C_2$ be the orthocenters of the triangles $BA_1C$, $CB_1A$ and $AC_1B$ respectively. Prove that:
a) triangles $ABC$ and $A_2B_2C_2$ have the same centroid;
b) the centroids of the triangles $AA_1A_2$, $BB_1B_2$, $CC_1C_2$ form a triangle similar with $ABC$.
2022 Bulgaria JBMO TST, 1
Determine all triples $(a,b,c)$ of real numbers such that
$$ (2a+1)^2 - 4b = (2b+1)^2 - 4c = (2c+1)^2 - 4a = 5. $$
2023 Tuymaada Olympiad, 8
Given is a positive integer $n$. Let $A$ be the set of points $x \in (0;1)$ such that $|x-\frac{p} {q}|>\frac{1}{n^3}$ for each rational fraction $\frac{p} {q}$ with denominator $q \leq n^2$. Prove that $A$ is a union of intervals with total length not exceeding $\frac{100}{n}$.
Proposed by Fedor Petrov